Решение:
\( 1,7 : \frac{(4,5 \cdot 1\frac{2}{3} + 3,75) \cdot \frac{7}{135}}{\frac{5}{9}} - (0,5 + \frac{1}{3} - \frac{5}{12}) = 1\frac{17}{84}\)

\(
1) 4,5 \cdot 1\frac{2}{3} = 4,5 \cdot \frac{5}{3} = 1,5 \cdot \frac{5}{1} = 7,5;
\)

\(
2) 7,5 + 3,75 = 11,25 = 11\frac{25}{100} = 11\frac{1}{4};
\)

\(
3) 11\frac{1}{4} \cdot \frac{7}{135} = \frac{45}{4} \cdot \frac{7}{135} = \frac{45 \cdot 7}{4 \cdot 135} = \frac{1 \cdot 7}{4 \cdot 3} = \frac{7}{12};
\)

\(
4) \frac{7}{12} : \frac{5}{9} = \frac{7}{12} \cdot \frac{9}{5} = \frac{7 \cdot 9}{12 \cdot 5} = \frac{7 \cdot 3}{4 \cdot 5} = \frac{21}{20} = 1\frac{1}{20};
\)

\(
5) 1,7 : 1\frac{1}{20} = 1\frac{7}{10} : \frac{21}{20} = \frac{17}{10} \cdot \frac{20}{21} = \frac{17 \cdot 20}{10 \cdot 21} = \frac{17 \cdot 2}{1 \cdot 21} = \frac{34}{21} = 1\frac{13}{21};
\)

\(
6) 0,5 + \frac{1}{3} = \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6};
\)

\(
7) \frac{5}{6} - \frac{5}{12} = \frac{10}{12} - \frac{5}{12} = \frac{5}{12};
\)

\(
8) 1\frac{13}{21} - \frac{5}{12} = 1\frac{52}{84} - \frac{35}{84} = 1\frac{17}{84}.
\)
Ответ: \(1\frac{17}{84}\).

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